exam set 2023

Question 1

Question 1.a

Instead of describing the outcome of this game, focus on one of the conditions for a stable coalition: When would a country join?

Answer 1

Answer:

First we rephrase somethings, note that e = E - Sum_i * a_i. Moreover, let a^c denote that abatement level that cooperation requires from all coalition´s members and by pi^c_i the corresponding indiviudal payoff. Denote by a^n and pi^n_i the non-cooperative abatement and payoff level respectively of the non-signatories.

Suppose that there are k signatories of the treaty. Their problem is to maximize, by choosing the abatement level a^c, their joint payoff

math

To determine the signs of the quadratic expression, we consider intervals aroudn the roots k = and k = 2

- When k < 0, the expression (2-k)k is negative (Also k can not be negative, when it represent then numbers of members). 
- When 0 < k < 2, the expression (2-k)k is positive. 
- When k > 2, the expression is negative.

This means that for it is beneficial for a country to join the coalition, if there is between 0 < k < 2 countries allready participating. In simpler terms, if there allready is 1, but not more than 1 when we think of countries as nondivideble, then other countries would like to join the coalition.

If thus result is stated with a weak inequality instead, we have

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The solution is then k = 0,1,2 , meaning that countries would like to join the coalition if there are zero, one or two mermbers in the coalition allready.

Question 2

Question 1.b)

Assume that each country has some disutility of being an outsider when there exists a coalition of emission abating countries, and that this disutility is larger the larger this coalition is. What could be reasons for such preferences?

Answer 2

Answer: For coalition formation, the disutility might result from penalizing free-riding. More specifically, the incentive for an indidvidual country to join a climate may be strengthened if the coalition can enforce a credible penalty for non-cooperative beahvior of outsider. Examples are trade sanctions and tariffs on import from outsiders (alternatives to a general tariff would be a carbob tariff or border carbon adjustment).

Question 3

Question 1.c)

Let this feature be captured by subtracting the term gamma k from the payoff of those countries that are not part of the coalition, where k is the number of countries that are part of the coalition, an gamma is a constat. Whitout solving for when a country would join a coallition, would more or fewer countries join the coallition compared to what you find in a) why

Answer 3

Answer: It does not affect the individually rational abatemnt levels for coalition members and outsiders. Equally the payoff to coallition members pi^c are not changed. However the payoff for an outsider pi^n has change, due to the increased cost of not cooperating. A likely effect of a penalty on outsiders would be, that more contries would like to join the coallition compared to in a)

Question 4

Problem 2: Policy instruments and uncertainty (weight 33%)

Consider an economy where profit maximizing producers of a private good emit a uniformly mixing
flow pollutant, which damages the environment.

Let 𝑀 = ∑𝑚𝑘 be aggregate emissions, where 𝑚𝑘 is firm 𝑘’s emissions.

Let the social benefits of pollution, 𝐵(𝑀), be given by the maximal private good production that is
possible for an aggregate emission level 𝑀, and let the social damage of pollution 𝐷(𝑀) be
determined by consumers’ valuation of the environmental quality, measured in units of the private
good.

An environmental regulator wants to maximize the net social benefits of pollution. To achieve this,
the regulator will cchoose one of two alternative policy instruments: a uniform emission tax or a
tradable permit.

Question 2.A)

Assume that the regulator knows the marginal social damages of pollution 𝐷′(𝑀). Due to
uncertainty on the abatement costs, the regulator only knows the slope of the marginal
social benefits of emissions:

𝐷′(𝑀) = 2𝑀 − 1
𝐵′(𝑀) = 𝑋 − 3𝑀

Discuss the differences in welfare loss due to the choice of a price instrument over a
quantity instrument and vice versa.

You can also use figures.

Can you recommend to the regulator an instrument choice (emission taxes or tradable
permits)?

Provide reasons for your answer.
How would your answer change if, instead, 𝐵′(𝑀) was given by 𝐵′(𝑀) = 𝑋 − 0.5𝑀?

Answer 4

Answer: This can be done in a figure with the 𝐷′(𝑀) and 𝐵′(𝑀) curves, where welfare losses are
illustrated by parallel shifts to the 𝐵′(𝑀) curve.

The key result is derived by comparing the (absolute value of the) slope of the 𝐵′(𝑀) curve is greater than the slope of the 𝐷′(𝑀) curve.
For the marginal damages, the slope is given by 𝐷′′(𝑀) = 2. For the case 𝐵′(𝑀) = 𝑋 − 3𝑀, we obtain 𝐵′′(𝑀) = −3. Because |𝐵′′(𝑀)| > 𝐷′′(𝑀), taxes are preferred to permits as
they lead to smaller efficiency losses.

For the case 𝐵′(𝑀) = 𝑋 − 0.5𝑀, we obtain 𝐵′′(𝑀) = −0.5. Because now instead|𝐵′′(𝑀)| < 𝐷′′(𝑀), permits are preferred to taxes as they lead to smaller efficiency losses.

Question 5

Question 2.B)

Assume now that the regulator estimates that 𝑋 = 10, and introduces an emission tax based on this assumption. However, the true value is 𝑋 = 8. What will optimal emissions and actual emissions be? Show graphically the loss in net benefits due to the regulator’s mistaken estimate of 𝑋.

Answer 5

Answer:

Emission tax regulator estimate

Inserting X = 10

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Find the optimal M for the regulators estimate,by setting the benefit function equal the damage function, hence

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Insert the optimal M in the benefit funvtion, so we can derive the Pigovian tax.

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We find the pigovian tax to be equal to the marginal benifit of pollution one more emission. The MB is equal to the marginal social cost, which is internalized with the a tax of the same size. In this way we corrects the market failure and allign private incentives with social wellfare.

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True value

Inserting X = 8

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We find the level of actual emissions, when we take the pigovian tax calutaed from the regulators estimate B´(M) = 8 - 3M and sets it equal to the pigovian tax, hence

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This is the level of the emissions using af tax calculated on a wrong esimate.

We now find the true optiamal level of emissions given X=8. We do the same as in the start, setting benefit and damage function equal, thus

math

Optimal emissions are higher 27/15 > 23/15. This means that we in the begining overestimated the level of emissions, which we calculated to be 33/15 from the estimated guess on X = 10. We can draw the efficiency loss from using a tax in contrast to using an emission permit scheme in figure 2.b.1. Due to the slope of the was greater, |𝐵′′(𝑀)| > 𝐷′′(𝑀), taxes are preferred to permits as they lead to smaller efficiency losses.

see figure in note 4

Question 6

Question 2.b

As before the regulator believes that 𝑋 = 10, while the true value is 8. In this case, the regulator introduces tradable permits based on the assumption that 𝑋 = 10. Show using a figure the loss in net benefits due the regulator’s mistaken estimate of 𝑋. Is this loss larger or smaller that the loss you found in point b)? Why?

Answer 6

Answer: The regulator will issue to many permits based on the assumption X = 10 while the true value is X = 8. This would increase the level emissions over the optimal point, which would have a greater social cost then using the tax. Graphically and qualitatively speaking, this corresponds to the red traiangle in the figure above, which represent the differences between the assumed and acutal MAC curve an emission level in figure 2.b.1

Question 7

Question 3.b

Assume now that the firms know the real value of 𝑋, so there is an information asymmetry between firms and the regulator.

If the regulator announces that it will use an emission tax, and then asks firms about their abatement costs, can the firms be expected to tell the truth? Why? Will they understate or overstate the real value?

What will happen if the regulators uses tradable permits? You can use figures to clarify your
answers.

Answer 7

Answer: If the regulator announces a tax, firms have an incentive to understate abatement cost, under the assumption that they only think about maximazing their profits and do not care about the negative externalitities of higher emissions. Firms benefit because they emmit more than if they told the truth (ans so incur lower real abatement costs). Also, the tax rate is lower than it would be otherwise.

If instead the regulater announces a permit scheme, the incentive is to overestimate these costs. Exaggerating abatement costs is better for firms than being truthful as more permits are issued, and so they incur lower real emission abatement costs, due to they are agin able to emit more then if they told the truth.